Two coils, one mounted on a stator and the other on a rotor,

Two coils, one mounted on a stator and the other on a rotor,

Two coils, one mounted on a stator and the other on a rotor, have self- and mutual inductances of L11 = 3.5mH L22 = 1.8mH L12 = 2.1cos ? mH where ? is the angle between the axes of the coils. The coils are connected in series and carry a current i = v2I sin wt
a. Derive an expression for the instantaneous torque T on the rotor as a function of the angular position ?.
b. Find an expression for the time-averaged torque T avg as a function of ?.
c. Compute the numerical value of T avg for I = 10 A and ? = 90°.
d. Sketch curves of T avg versus ? for currents I = 5, 7.07, and 10 A.
e. A helical restraining spring which tends to hold the rotor at ? = 90° is now attached to the rotor. The restraining torque of the spring is proportional to the angular deflection from ? = 90° and is – 0.1 N · m when the rotor is turned to ? = 0°. Show on the curves of part (d) how you could find the angular position of the rotor-plus-spring combination for coil currents I = 5, 7.07, and 10 A. From your curves, estimate the rotor angle for each of these currents.
f. Write a MATLAB script to plot the angular position of the rotor as a function of rms current for 0 < i=””>< 10=”” a.=”” (note=”” that=”” this=”” problem=”” illustrates=”” the=”” principles=”” of=”” the=”” dynamometer-type=”” ac=””>

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